# Wireless Charging & Cooktop Simulator - Mutual Inductance, Coupling, Efficiency

> Compute coil mutual inductance and coupling k, simulate resonant wireless power and frequency splitting, align a Qi phone, see why hobs need iron pans.

- URL: https://qilingtools.com/en/physics/wireless-power
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Mutual inductance and coupling between two coils are computed exactly with elliptic integrals and the Neumann formula, and you can drag the receive coil in a side view to watch the alternating field lines and induced EMF change with distance, offset and tilt. The resonant-link tab gives series-series efficiency, the optimal load and frequency splitting when over-coupled. The Qi tab lets you drag a phone to see efficiency drop when misaligned, and the induction-cooktop tab estimates pan heating from skin depth and surface resistance, explaining why iron works and aluminium does not.

## How to use

1. On Coil coupling, pick a preset or set the radii, turns and spacing, then drag the receive coil and watch k change
2. On Resonant link efficiency, set frequency, Q values and load to see the efficiency curve and the frequency split
3. On Qi phone charger, drag the phone to compare coupling and efficiency aligned and offset
4. On Induction cooktop, choose the pan material, frequency, diameter and thickness to see skin depth and heating power

## Features

- Exact elliptic-integral mutual inductance for coaxial coils; Neumann double integral for offset and tilt
- Draggable side view with field lines animated over the AC cycle and an induced-EMF bar
- Series-series resonant link: maximum efficiency, optimal load and a phasor-solved frequency-splitting curve
- Top view of a Qi pad: drag the phone and get k, efficiency and relative output live
- Induction hob: skin depth and thickness-corrected surface resistance for five pan materials, heating power integrated over the pan
- Notes on RFID/NFC read range by frequency band

## FAQ

### Why does wireless charging only work over such short distances?

It relies on near-field magnetic coupling. Beyond about one coil radius the transmit field falls roughly as 1/d³, and the coupling coefficient k drops just as fast. A Qi coil is only about 4 cm across: a few millimetres apart k is 0.3–0.6, and at 2 cm there is little left. Going further needs bigger coils, or high-Q resonance so that a small k still gives decent efficiency.

### How can resonance be efficient when k is tiny?

With series-series compensation the coil reactance is cancelled and efficiency depends only on the figure of merit U = k√(Q₁Q₂). The maximum efficiency is η = U²/(1+√(1+U²))². For k = 0.05 and Q₁ = Q₂ = 300, U = 15 and efficiency is still above 87%. That needs the load matched to R₂√(1+U²); the tool shows how efficiency falls when it is not.

### What is frequency splitting?

When two resonators tuned to the same frequency are coupled strongly (k above roughly 1/Q), the system gets two new natural frequencies near f₀/√(1±k). Load power versus frequency changes from one peak to two, and the original f₀ becomes a dip. So a charger fixed at f₀ delivers less power when the coils are too close, unless it tracks frequency or adapts the load.

### Why can’t induction hobs heat aluminium or copper pans?

Heating power is proportional to the surface resistance R_s = √(πfμρ). Iron has high permeability and fairly high resistivity, so at 25 kHz its skin depth is under 0.1 mm and R_s is large. Aluminium and copper are non-magnetic with low resistivity, so their R_s is only a few percent of iron’s: they barely warm at the same current, and the coil sees too small a load for the inverter to match. Matching iron would take hundreds of times the frequency, which is why only special all-metal hobs handle them.

### How accurate are the inductance calculations?

Coaxial single-turn mutual inductance uses the closed form with complete elliptic integrals K and E, accurate to floating point. Offset and tilt use the Neumann formula integrated numerically over 160×160 segments, which agrees with the closed form to well under 1%. Multi-turn coils are treated as wound at a single radius and self-inductance uses μ₀a[ln(8a/r)−2], so real flat spiral coils will differ somewhat. Treat the results as estimates for learning and early design.

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